Non-connected Lie groups, twisted equivariant bundles and coverings

Let $\Gamma$ be a finite group acting on a Lie group $G$. We consider a class of group extensions $1 \to G \to \hat{G} \to \Gamma \to 1$ defined by this action and a $2$-cocycle of $\Gamma$ with values in the centre of $G$. We establish and study a correspondence between $\hat{G}$-bundles on a manifold and twisted $\Gamma$-equivariant bundles with structure group $G$ on a suitable Galois $\Gamma$-covering of the manifold. We also describe this correspondence in terms of non-abelian cohomology. Our results apply, in particular, to the case of a compact or reductive complex Lie group $\hat{G}$, since such a group is always isomorphic to an extension as above, where $G$ is the connected component of the identity and $\Gamma$ is the group of connected components of $\hat{G}$.